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The "Vernier scale" is an ingenious device to make a "simple" scale readable with greater precision.
You can't have science without measuring instruments
Of course, if money and pride are no object, you can just buy your measuring instruments. But where's the fun in that? A lot of fun can be had trying to walk in the footsteps of the first scientists, people who couldn't just buy the instruments they were inventing, so science could move ahead.
The Vernier scale crops up on many instruments. (Not least theolodites. The page you are reading arose because I wrote elsewhere about making your own theolodite, with Vernier scale.)
Some of the details of what one is, and all of the details of how to use one are covered elsewhere. This page is just about MAKING one.
Imagine you are making a pair of the sort of calipers (WP) which look like....
(My thanks to a Wikipedia artist for that.)
In a simple one, there's a scale on the "spine" of the caliper, and a pointer attached to the sliding "jaw". Where the pointer points on the scale tells you how far open they are. (In actual fact, this sort of caliper almost always uses a Vernier scale, but bear with me? Pretend we have one that doesn't?)
Further imagine that the scale is divided down to millimeters... one mark per millimeter, nothing in between those marks. The marks, of course, must be as close to exactly 1mm apart as possible.
So far so good? Just setting the scene, so far.
Remember: All I am talking about here is how you can MAKE a Vernier scale. Why you would, how you would use is: See other! (But... having said that, partly because it may help with what comes next, here is a main scale, with a Vernier scale shown beside it. This is three diagrams in one. You would have just one Vernier, and the two move relative to one another, in response to what you are measuring. The three Verniers are showing 0.24, 1.52 and 2.97.
What you need, to make a Vernier scale, is a set of marks which are very slightly less widely spaced than the marks on the main scale.
If you want to enjoy all the benefits of decimal numbers, you need to make something with ELEVEN marks spread over (as exactly as you can make it) the same distance as TEN marks occupy on the main scale. (These two spans, though the "magic" of what programmers call "the fencepost problem", would be 9mm and 10 mm long, respectively.) (Just to make explaining all of this a little harder, you won't actually use the 11th mark on the "squashed thing", once you have made it. You only need it long enough to make sure that the squashed thing is the right length.)
Remember: The fun is in doing it yourself, with things you already have, hopefully things that aren't clever or expensive.
I suggest you pause your reading here. Go away for a day or so. Can you think of a good way to draw nine lines in the distance occupied by ten on the main scale. Much more fun than using someone else's ideas.
(Here there is a gap, to spare you accidentally seeing my ideas before you are ready. There IS more... further down the page....
... but not for NOW...
... for later... when you are ready...
... for when you've tried to solve the problem yourself!...
... okay... here we go...
Both ideas work on the principle of "measuring big"/ "using small". In both cases, my illustration will be of trying to draw 11 fine lines very precisely spread across a 10 millimeter spread. (The number of lines, and the span might be different in your wants. The techniques can be adapted, as necessary.)
Purely mechanical. Needs good ruler, big sheet of paper, skill with fine pen. You'll have to fiddle with the numbers given, to "fit" the size of the sheet of paper you have, and the length of your straight edge. The smaller you do the diagram, the greater your errors will be, other things being equal. (I'd be grateful for feedback, quoting "Page "mm1t1gv.htm", telling me ACTUAL numbers that work well for 11 lines in 1 cm.)
Place eleven dots along the edge of the page, 20mm apart.
About 100 mm from first line, do a second line of eleven dots... 18mm apart.
Connect the pairs of dots... first in first line to first in second, 2nd in 1st to 2nd in 2nd, etc... and extend the lines, a long way below the second line of dots. The lines that will produce should all get closer and closer to one another, the further you get from the two lines of dots.
If you got "the numbers" right, at some point, there will be 11 lines spanning 1cm. Carefully trim the paper, parallel to your lines, at "the right place", and you will have what we wanted. Note: If you are slightly off (parallel to the central line), the error in your scale will be slight, won't it? I LIKE construction techniques which are forgiving, don't you?
We can't eliminate error... but we can do things to keep it smaller than it would otherwise be.
The trick here was that we avoided trying to measure intervals of 0.90909 mm. Quite hard to do, I would imagine. If our "20mm apart" lines are out by 5%, then our "11 per 10cm" lines will be out by something like 0.045mm... a lot better than I could do the job directly! (You might want to look at my estimates of the error... Doing such estimates is another important enterprise in the world of science. Were my estimates right? Not the "out by 5%"... the "IF out by 5%, lines out by 0.045mm".) (And yes, I have maltreated the use of significant digits atrociously.)
This can be implemented in many ways. I hope I've expressed it in a way that makes that obvious.
Carefully, SOMEHOW, draw (or obtain) eleven parallel lines a certain distance apart... as much more than the final 0.9mm as you can stand. (The farther apart they are, the better... and the more "work" you will have.)
You can draw them "literally", i.e. with pen and ruler, or generate them in a drawing program (Libre / OpenOffice's wordprocessor COULD be used, but if you have the suite installed either the spreadsheet, or, Do Ya Think?, the drawing package.
Also draw a "long" line, on the same artwork.
Measure the spread of the 11 lines, overall. Let's say they come to 5cm. Measure the long line. Let's say it comes to 10cm.
Reprint (use photocopier's "zoom", if nothing more subtle comes to mind!) the artwork, shrinking it down to the point that the long (10cm line) is now 2cm long. The 11 lines will now span 1cm, won't they?!
Give this approach some thought. See why it gives you a way to generate the 11 lines well? What error would you have to concede might be present if you used the numbers above, and had to concede that the "10" cm line might have been anything from 9.9cm to 10.1cm?
AFTER I'd written up the ideas above, I had the following idea... and originally nearly dismissed it as not being suitable. (I had to do this whole page twice, as it happens... I'd "flipped over" in my mind what we want to do. Originally, I thought we needed to SPREAD the lines, so that 10 took the same span as 11. If THAT's what we ACTUALLY wanted, this idea would work! But we want to SQUASH the line spacing, so it won't. Sigh. But here it is, if ever you want to SPREAD some lines!...
If you will allow yourself a piece of good quality graph paper, ruled to 1mm... and even "pretty good" paper, in our society, is readily accessible (affordable!)... then here's what you do...
Cut a piece diagonally. The steeper the cut, the more the lines at the edge of the paper will be "spread out". Of course, for extra credit, you could calculate the correct angle to cut at, for a given percentage "spreading". Sadly, the lines won't be perpendicular to the edge of the piece of paper, which might lead users to misread the scale... but probably, at least, in a consistent manner. But Really Easy To Make! (Every solution has pros and cons.)
What's YOUR solution! (Answers on the back of either a $5 or £5 note especially welcome! Or as posts at Facebook, with a citation of this page!)
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Do it yourself: Make theolodite scale and pointerSearch across all my sites with the Google search button at the top of the page the link will take you to.
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Page tested for compliance with INDUSTRY (not MS-only) standards, using the free, publicly accessible validator at validator.w3.org. Mostly passes. There were two "unknown attributes" in Google+ button code, and three in the Flattr. Sigh.